
where \(\mathbf {x}_s=\lbrace x_{s,1},x_{s,2}\rbrace \) . The optimal solution of () is always binary, i.e., \(x_{s,1}^*,x_{s,2}^*\in \lbrace 0,1\rbrace \)  [1]}, [2]}. Similar to (REF ), (REF ) is concave. Therefore, we consider a quadratic upper-bound [3]} for it as follows:
\(&\left\Vert \mathbf {x}_s+\epsilon \mathbf {1}\right\Vert _q^q\le \left\Vert \underline{\mathbf {x}}_s^i+\epsilon \mathbf {1}\right\Vert _q^q+q\big \langle \left\lbrace (\underline{x}_{s,1}^{i}+\epsilon )^{q-1},(\underline{x}_{s,2}^{i}+\epsilon )^{q-1}\right\rbrace \nonumber \\&, (\mathbf {x}_s-\underline{\mathbf {x}}_s^i)\big \rangle +\frac{e}{2}\left\Vert \mathbf {x}_s-\underline{\mathbf {x}}_s^i\right\Vert _2^2,\hspace{8.5359pt}\forall s.\) 
